Some people consider all elementary particles to be non-existent. They tell us that they are changes in the field configurations that last which are called "real particles", while those that not last long enough are called "virtual particles".
But field configurations refer to wavefunctions that are a part of the entire field. Seen in this light, particles do exist, with an associated wavefunction. And so do virtual particles, though they don't last long enough to be "real". They find themselves (only by saying this already gives some "realness" to their existence) not on mass-shell and appear in the interaction between real particles, which do find themselves on mass-shell (or in the field configuration of the vacuum).
In the interaction case (as well as in the vacuum field configuration case or, for that matter, even in the case of changing field configurations, representing real particles), they are taught to be just mathematical constructions.
One can read [here] (a nice exchange of views, questions, and answers):
However, it appears to be a very good question whether virtual particles actually "exist" i.e. if they play a causal role. There is the common talk about virtual particles being exchanged and thus mediating interactions. This view was criticized e.g. by Mario Bunge (1970) in his „Virtual Processes and Virtual Particles: Real or Fictitious?”, International Journal for Theoretical Physics, Vol. 3, No. 6, 507-508. A rather forceful argument to the same effect was given by Weingard in his „Virtual Particles and the Interpretation of Quantum Field Theory“, in: H. R. Brown und R. Harré (Hrsg.) „Philosophical Foundations of Quantum Field Theory” 1988, Oxford: OUP. Weingard's bottom line is, that virtual particles are an artifact of perturbation theory only.
It's also possible to execute a Feynman diagram without the use of virtual particles, i.e., without the use of particles that not lay on their mass shell, but I didn't take a "deep" look into that possibility. It can be found here.
My question is very simple: If it is said that virtual particles are just mathematical constructs (just as real particles), does that imply that there are no real correspondences with the real world? It seems to me that something real must happen in the reaction between two particles. Or that elementary particles must be real particles and not mathematical constructions (changes in field configurations). As I said, changes in the field configurations correspond to wavefunctions which in turn correspond to real particles. These can be point-like or have some other form (I'm not referring to string theory).